In the same way you can project a [[Vector]] onto another [[Vector]] in the traditional sense, you can project any abstract vector in a [[Vector Space]] to another with similar reasoning, replacing [[Vector Magnitude|magnitude]] and the [[Dot Product]] with an [[Inner Product]]. $\huge \proj{\vec v}{\pa{\vec u}} = \frac{\braket{\vec v,\vec u }}{\braket{\vec v, \vec v}} \vec v $ In [[Functional Analysis]], this can be used with techniques like the [[Gram Schmitt Process]] to create a [[Basis]] from [[Function|Functions]]. Typically this means having a basis constructed from an infinite amount of basis functions, which is how [[Fourier Transform]] operates. In that case, it uses an infinite amount of [[Orthogonal]] [[Sinusoidal Functions]] >[!example] >$\huge > \vec a >f(x) = \sum^{\infty}_{n} \langle \rangle > >$